Subgroups — #2112

Order 128.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×M₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂³×C₄	C₄	1	1	yes	characteristic
5	C₂²×C₈	—	2	4	no	—
6	C₂²×C₈	C₄	1	1	yes	characteristic
7	C₂²×C₈	C₄	1	2	yes	—
8	C₂²×C₈	C₂²	1	2	yes	—
9	C₂²×C₈	C₂²	1	1	yes	characteristic
10	M₃₂	—	2	4	no	—
11	C₂×C₁₆	C₂²	1	4	yes	complemented
12	C₂⁴	C₈	1	1	yes	characteristic
13	C₂²×C₄	—	2	2	no	—
14	C₂²×C₄	—	2	2	no	—
15	C₂²×C₄	C₈	1	1	yes	characteristic
16	C₂²×C₄	C₈	1	2	yes	—
17	C₂²×C₄	C₂×C₄	1	2	yes	—
18	C₂²×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈	—	2	2	no	—
20	C₂×C₈	—	2	4	no	—
21	C₂×C₈	—	2	4	no	—
22	C₂×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈	D₄	1	4	yes	—
25	C₂×C₈	C₂³	1	1	yes	characteristic
26	C₁₆	—	2	4	no	—
27	C₂³	—	2	2	no	—
28	C₂³	—	2	4	no	—
29	C₂³	C₂×C₈	1	1	yes	characteristic
30	C₂³	C₂×C₈	1	2	yes	—
31	C₂×C₄	—	2	2	no	—
32	C₂×C₄	—	2	4	no	—
33	C₂×C₄	—	2	2	no	—
34	C₂×C₄	—	2	2	no	—
35	C₂×C₄	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄	C₂×C₈	1	2	yes	—
37	C₂×C₄	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄	M₁₆	1	2	yes	—
39	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₈	—	2	2	no	—
41	C₈	C₂²⋊C₄	1	2	yes	—
42	C₈	C₂×D₄	1	2	yes	—
43	C₂²	—	2	8	no	—
44	C₂²	—	2	2	no	—
45	C₂²	—	2	2	no	—
46	C₂²	—	2	2	no	—
47	C₂²	C₂²⋊C₈	1	2	yes	—
48	C₂²	M₃₂	1	4	yes	complemented
49	C₂²	C₂²×C₈	1	1	yes	characteristic
50	C₄	—	2	2	no	—
51	C₄	C₂²⋊C₈	1	2	yes	—
52	C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄	C₂×M₁₆	1	1	yes	characteristic
54	C₂	—	2	2	no	—
55	C₂	—	2	4	no	—
56	C₂	C₂×C₂²⋊C₈	1	1	yes	characteristic
57	C₂	C₂×M₃₂	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂³
commutator, socle	C₂²	C₂²×C₈
prime radical	C₂⁴	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×M₃₂	C₂	semidirect
3. C₂²⋊C₁₆	C₂	semidirect
11. C₂×C₁₆	C₂²	semidirect
48. C₂²	M₃₂	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₈	2
C₂×M₃₂	2
C₂²⋊C₁₆	2